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Chapter 6: Problem 91

An infinitely long, solid, vertical cylinder of radius \(R\) is located in an infinite mass of an incompressible fluid. Start with the Navier-Stokes equation in the \(\theta\) direction and derive an expression for the velocity distribution for the steady-flow case in which the cylinder is rotating about a fixed axis with a constant angular velocity \(\omega\). You need not consider body forces. Assume that the flow is axisymmetric and the fluid is at rest at infinity.

Short Answer

Expert verified
The velocity distribution of the flow around an infinitely long rotating cylinder is \( v_\theta(r) = \omega R^2 / r \).

Step by step solution

01

Write down the Navier-Stokes equation in the θ direction

The Navier-Stokes equation in cylindrical coordinates \( (r, \theta, z) \) in the \( \theta \) direction, neglecting body forces, is given by: \[ 0 = -\frac{1}{r}\frac{\partial p}{\partial \theta} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}(r \frac{\partial v_\theta}{\partial r}) + \frac{1}{r^2}\frac{\partial^2 v_\theta}{\partial z^2} - \frac{v_\theta}{r^2}\right] \] Here, \( p \) is the pressure, \( \mu \) is the dynamic viscosity and \( v_\theta \) is the velocity in the \( \theta \) direction.
02

Simplify the Navier-Stokes equation for the given assumptions

Given the axisymmetric and steady flow, the flow does not change with angle \( \theta \) or height \( z \), so all derivatives with respect to \( \theta \) and \( z \) vanish. Similarly, there is no motion in radial \( r \) direction, hence the radial component of velocity is zero. Thus, the simplification of the Navier-Stokes equation leads us to: \[ 0 = \mu \left[\frac{1}{r}\frac{\partial}{\partial r}(r \frac{\partial v_\theta}{\partial r}) - \frac{v_\theta}{r^2}\right] \]
03

Integrate to derive the velocity distribution

Integrating the equation above twice with respect to \( r \), we obtain the velocity profile in the \( \theta \) direction: \[ v_\theta(r) = A\ln(r) + B \] where \( A \) and \( B \) are constants to be determined by the boundary conditions.
04

Apply the boundary conditions

The boundary conditions for this problem are: at the surface of the cylinder \( r=R \), the fluid has the same angular velocity as the cylinder \( v_\theta(R) = R \omega \) and at \( r \rightarrow \infty \), the fluid is at rest \( v_\theta = 0 \). Applying these conditions, we can solve for the constants \( A \) and \( B \). This gives us: \[ v_\theta(r) = \omega R^2 / r \] This is the velocity distribution around an infinitely long rotating cylinder.

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